Random Effects Models in ANOVA: Understanding Random Effects and F Tests, Study notes of Statistics

Download Random Effects Models in ANOVA: Understanding Random Effects and F Tests and more Study notes Statistics in PDF only on Docsity! 1 Random Effects models Chapter 11 begins with an example of studying variation between cardboard carton machines. Out of 50 machines, 10 are randomly chosen for study. We wish to apply the conclusions about these 10 machines to all 50 machines, so the chosen groups (effects) are random effects. The model for random effects ANOVA from a completely randomized design looks familiar: Yij = µ + αi + εij , where µ is the population grand mean, and now the αi terms are random, having normal distributions with mean zero and variance σ2α. The random effects αi are independent of the errors εij . The null hypothesis of interest in this experiment is H0 : σ2α = 0, implying that all αi terms are zero. Some other changes with random effects models: i) We are generally not interested in multiple comparison tests, ii) It is of interest to test for main effects even in the presence of interactions, and iii) expected values of mean squares change, which for more complicated models can lead to F tests with denominators other than MSE . 1.1 Constructing F tests As we have previously mentioned, the goal of an F test is to arrange it so that under H0, F ≈ 1, and under HA, F >1. Display 11.1 in our text shows that for the completely randomized design with random effects, the usual F = MSTREAT /MSE satisfies this criteria. Display 11.2, however, shows that for a two-factor random effects model, the test for A would be F = MSA/MSAB in order to satisfy these criteria. Display 11.3 shows an even stranger situation, where there is no term that can be used in the test for A that satisfies these criteria. We will soon learn how to calculate these expected values of mean squares (EMS’s) and what to do in situations like in Display 11.3 when an exact F test is not available. Chapter 11 in our text discusses methods for estimating variance components and confidence intervals for variance components, using a method known as the method of moments (otherwise known as the ANOVA method). We will instead use other methods such as the method of maximum likelihood, which is available in SAS Proc MIXED. 1